Just And Mean-Tone Tunings
The octave is universally recognized as the most natural musical interval other than the unison. Traditionally, the division of the octave into smaller intervals was made with frequency ratios of small integers (called "just" intervals) so that harmonic relationships between the notes could be achieved. It was recognized that a scale composed entirely of just intervals had inevitable pitch errors because concatenated just intervals do not form an exact octave. With fixed-pitch instruments, various tunings evolved, in which the residual errors, known as commas, were lumped onto different intervals of the musical scale. Mean-tone tuning was invented to distribute the comma onto two adjacent intervals, such that neither interval had a large amount of error compared to their "just" counterparts.
Tempered Tunings
The frequency chosen to begin and end the scale defines the key of a musical expression. The key, in turn, defines the frequencies of the set of notes within the scale. Over the most recent several centuries, transitions between multiple keys within the same piece of music became a prominent feature of music. The necessity to play notes from all keys on demand presented special challenges in tuning the instruments, because, on most common instruments with fixed tuning, such as 12-tone keyboards, near-harmonious tunings, such as the traditional mean tone tuning, could not be achieved in multiple keys simultaneously. This lead to various compromised tunings and the concept of "tempering" the division of the octave to facilitate transposing between keys without re-tuning. Mean-tone tuning, which can be considered a tempered scale itself, found its ultimate expression in equal temperament, in which the octave is divided into intervals that are exactly equal to one another. With the equal-tempered scale, the comma is spread onto all intervals of the octave.
It should be noted that the equal-tempered scale compromises the harmony found in "just" and mean-tone tunings in favor of the freedom to change keys. But because music in Western cultures has continued to evolve within this scale, and influenced other cultures as well, key transpositions have become a necessary part of a significant musical heritage. As a result, a contemporary musical instrument must be able to produce, as accurately as possible, the 12-tone equal-tempered scale.
Equal Temperament and Geometric Series
A series of numbers in which each number is a constant multiple of the previous number is called a geometric series and the constant is called the geometric constant. The frequencies of the descending 12-tone equal-tempered scale are comprised of a geometric series with a geometric constant k whose value is ##EQU1##
Here, the number 2 represents the octave ratio, and 12 is the number of intervals within the octave. If truncated after 4 significant digits, this constant yields the decimal value k=0.9438.
The Rule Of 18
A common practice in the manufacture of the neck of a guitar is known as "the rule of 18". This rule requires that starting with the first fret from the nut, each fret be placed at 17/18 of the previous fret's distance to the bridge. As a consequence, the vibrating lengths of a string being fretted at successive frets comprise a geometric series with a geometric constant of 17/18. The decimal equivalent of the fraction 17/18, accurate to 4 significant digits, is 0.9444. This value is close to the value k within approximately 0.06%. In other words, the rule of 18 divides the neck of a musical instrument with nearly the same relationship as the frequencies of the 12-tone equal tempered scale.
U.S. Pat. No. 2,649,828 by Maccaferri, U.S. Pat. No. 4,132,143 by Stone, and U.S. Pat. No. 5,600,079 by Feiten make references to the inaccuracy of the fraction 17/18. Maccaferri and Feiten give accurate decimal values for k.
Whether the fraction 17/18 or a more accurate value of k is used, when building a guitar neck prior to the present invention, the frets had to be located with respect to scale length, but without respect to any other dimensions of the guitar or physical properties of strings.
With modern manufacturing it is not necessary to cut fret grooves one at a time or to calculate one fret's location from measurements on another. On a guitar neck that is divided conventionally, with the geometric constant k, the distance of any fret from the bridge can be represented for all frets by the single mathematical expression ##EQU2## In equation (1) n is the fret number, and L.sub.n is the active length of the string (distance from active fret to bridge). L.sub.0 (distance between nut and bridge) is defined as the scale length of the instrument. Equation (1) defines the location of all frets on an instrument correctly built according to the prior-art technique of geometric neck division. Referring to the geometric constant of equal temperament, note that Equation (1) can also be written as EQU L.sub.n =L.sub.0 .multidot.k.sup.n
This equation that defines fret locations of a conventional guitar contrasts with the equation that defines fret locations according to the present invention, in that the latter equation contains additional terms. These additional terms relate to string properties.